Lower bound of decay rate for higher-order derivatives of solution to the compressible fluid models of Korteweg type

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Introduction
In this paper, we are concerned with the lower bounds of decay rate for the global solution to the compressible Navier-Stokes-Korteweg system in three-dimensional whole space: ρ t + div(ρu) = 0, (ρu) t + div(ρu ⊗ u) − μΔu − (μ + ν)∇divu + ∇P (ρ) = κρ∇Δρ, (1.1) where t ≥ 0 is time, x ∈ R 3 is spatial coordinate and the unknown functions ρ = ρ(x, t) and u = (u 1 , u 2 , u 3 )(x, t) represent density and velocity, respectively. The pressure P (ρ) is a smooth function in a neighborhood of 1 with P (1) > 0. The constant viscosity coefficients μ and ν satisfy the following physical conditions: μ > 0, 2μ + 3ν ≥ 0. The constant capillary coefficient κ satisfies κ > 0. To complete the system (1.1), the initial data are given by (ρ, u)(x, t)| t=0 = (ρ 0 (x), u 0 (x)). (1.2) Furthermore, as the space variable tends to infinity, we assume Korteweg-type models, supposing that the energy of the fluid depends on standard variables and the gradient of the density, are based on an extended version of nonequilibrium thermodynamics. The Navier-Stokes-Korteweg model, describing the dynamics of a liquid-vapor mixture with diffuse interphase and being used to model the motion of compressible fluid with capillary effect of materiel, originals from the work of van der Waals [23] and Korteweg [17], and the modern form was derived by Dunn and Serrin [8]. It should be noted that the system (1.1) will reduce to the well-known compressible Navier-Stokes system if the capillary coefficient satisfies κ = 0.
There are many studies on the well-posedness of solutions to the compressible fluid models of Korteweg type. For the one-dimensional case, many researchers have studied extensively; refer to [3,5] and the references therein. Charve and Haspot [3] obtained the global strong solution in the case of Saint-Venant viscosity coefficients. Chen et al. [5] obtained the global existence of classical solutions with large initial data away from vacuum for the isothermal compressible fluid of Korteweg type under the condition that the viscosity coefficient and capillarity coefficient are dependent on the density. For the multidimensional case, Hattori and Li [12] showed the local existence of smooth solution with large initial data for the isothermal compressible fluid models of Korteweg type in R 2 . Later, Hattori and Li [13] showed the global existence of smooth solution with small initial perturbation for the isothermal compressible fluid models of Korteweg type in high dimensions in some Sobolev space. Danchin and Desjardins in [7] showed the existence and uniqueness of suitably smooth solutions in critical Besov space. This result was improved by Haspot [11] by showing the global existence of weak solution, while the initial data belong to the energy space. Bresch et al. [2] obtained global weak solutions for the isothermal Korteweg model in a periodic or strip domain without smallness of assumption on the initial data when the viscosity μ and the capillarity coefficient κ are dependent on the density. For more results about the well-posedness result, the readers can refer to [1,4,6,[14][15][16]18,20,21] and the references therein.
The study for the asymptotic behavior of solution to the compressible Navier-Stokes-Korteweg equations has attracted many scholars' attention. First of all, the researchers in [20,24,25] established the time decay rates for the global solution under the H N (N ≥ 3), H 2 and H 1 framework, respectively. More precisely, Wang and Tan [25] established the global existence of solution and built the time convergence rates for the case k = 0, 1, (1.4) Later, Gao et al. [10] proved that the convergence rates (1.4) come true when k ∈ [0, N] with N ≥ 3, which will be showed in Theorem 1.1. In order to obtain fast time convergence rates for the higher-order spatial derivatives of solution, Tan and Zhang [22] considered the case of initial data belonging to some negative Sobolev space rather than general L 1 space. More precisely, if the initial data (ρ 0 − 1, u 0 ) ∈ (H N +1 ∩Ḣ −s ) × (H N ∩Ḣ −s ) (N ≥ 3 and s ∈ 0, 3 2 ), they established time decay rates where k = 0, 1, 2, . . . , N − 1. On the other hand, Li [18], Wang and Wang [24] studied the time decay rates of smooth solution and strong solutions under the smallness assumption on the potential external force in some Sobolev space, respectively. It should be noted that the decay rate (1.4) is called "optimal" in the sense that this rate of solution for the nonlinear part coincide with the decay rate of linearized one. Thus, the aim of this paper is devoted to providing lower bounds of decay rate (coincide with upper rate) for the global solution itself and its derivatives. In other words, this implies that the decay rate (1.4) obtained in [25] is really optimal. Notation In this paper, the symbol ∇ k with an integer k ≥ 0 stands for the usual any spatial derivatives of order k. For example, we define . We also denote the Fourier transform F(f ) :=f . Denote by Λ s the pseudo-differential operator defined by Λ s f = F −1 (|ξ| sf (ξ)). Denote L 2 (R 3 ) and H s (R 3 ) as the usual Lebesgue space and Sobolev space. For the sake of simplicity, we write f dx : First of all, we recall the main results obtained in [10,25] in the following: Theorem 1.1. [10,25] Assume that the initial data ρ 0 − 1 ∈ H N +1 and u 0 ∈ H N for any integer N ≥ 3 and there exists a small constant δ > 0 such that
Next, we establish the lower bound of time decay rates for the global solution of (1.1)-(1.3) only under the H 4 × H 3 framework for the sake of simplicity. (1.9) Here, t * is a positive large time, and c 1 and C 1 are two positive constants independent of time.
Here, t * is a positive large time, and c 1 and C 1 are two positive constants independent of time.

Remark 1.3.
It should be pointed out that under the H 3 -framework, the decay rates (1.9) and (1.10) for the global solution to the compressible Navier-Stokes equations can only be obtained under the condition that k = 0, 1 (see [9]). However, these decay rates for the solution to the Navier-Stokes-Korteweg equations can be obtained when k = 0, 1, 2, 3. The difference here is the appearance of Korteweg term κρ∇Δρ that will obtain enough dissipation for the density.
Finally, we will establish the lower bounds of decay rates for the time derivatives of solution to the compressible Navier-Stokes-Korteweg system (1.1).
Here, t * is a positive large time, and c 2 and C 2 are two positive constants independent of time.
Remark 1.4. The lower bounds of decay rates for the time derivatives of density and velocity for the compressible Navier-Stokes-Korteweg system in the L 2 norm are obtained for the first time.
Now we make comments on the analysis of this paper. At first, we give the lower bound of decay rate for the higher-order spatial derivative of solution to the compressible Navier-Stokes-Korteweg equations (1.1). Define U and U l as the solution to the nonlinear and linearized problems, respectively. Define U δ := U − U l , then we have for any integer k For α l,k < α δ,k , if the solutions U l and U δ satisfy where C l,k and C δ,k are positive constants independent of time. For large time t, it holds on where C is a positive constant that independent of time. It is easy to obtain the lower bound of decay rate for the linearized part by applying the spectral analysis to the semigroup for the linearized Navier-Stokes  [19] to obtain the upper bound of decay rate for ∇ k U δ . Next, the upper and lower bounds of decay rate for the time derivative of velocity can be obtained by using the equation and lower bound of first-order spatial derivative. If we use the transport equation to obtain the lower bound of decay rate for the time derivative of density, we need to get the lower bound for the quantity divergence of velocity(i.e., divu). To achieve this target, we need to assume the smallness for the initial velocity in L 1 .
The rest of this paper is organized as follows. In Sect. 2, we establish the lower bound of decay rate for the solution itself and derivative, and then, we establish the upper and lower bounds of decay rate for the time derivatives of solution. In Sect. 3, we prove technical estimates used in Sect. 2.

Lower bounds of decay for spatial derivative
In this section, we will address the lower bound of decay rates for the solution itself and its derivative. To this end, the upper decay rates for the difference between the nonlinear and linearized parts will be established. Finally, we address the upper decay rate of solution for the higher-order spatial derivative.

Lower bounds of decay for spatial derivative
In this subsection, we will establish optimal time decay rates of solution for the compressible Navier-Stokes-Korteweg equations (1.1)-(1.3). Let us denote := ρ − 1, m := ρu. For simplicity, we take P (1) = 1; then, we rewrite (1.1) in the perturbation form as follows: where the function S = S( , u) is defined as The initial data are given as In order to obtain the lower decay estimate, we need to analysis the linearized part: with the initial data Here, we assume initial data for the linearized system (2.4) is the same as the original problem (2.1).
where c is a positive constant independent of time t.
Proof. By virtue of the semigroup theory for evolution equation, the solution ( l , m l ) of the linearized problem (2.4), (2.5) can be expressed by Applying Fourier transform to Eq. (2.7), it holds on (2.10) Then by computation, we obtain the Fourier transformĜ(ξ, t) of Green's function G(x, t) = e tB as follows:Ĝ where Thus, we can obtain the expression forˆ l (ξ, t) andm l (ξ, t) as follows, Then it is easy to verify that where η is a small but fixed positive constant. Due to the fact that Employing the mean value formula, we get cos(|ξ| + O(|ξ| 3 )) = cos(|ξ|t) + O(|ξ| 3 t); then, we have which, together with the above inequality, we obtain that therefore, we only need to consider the lower bound of I 1 . We claim that the following estimate (which will be proved in Sect. 3), Hence, we can easily obtain that for s = 0, 1, 2, 3, where C is a positive constant independent of time. Similarly, we can also derive that for s = 0, 1, 2, 3, with C a positive constant independent of time. Therefore, we finish the proof of this proposition.
In order to obtain the lower bound for the solution of the compressible Navier-Stokes-Korteweg equation (2.1), we need to address the upper decay rate for the difference between the nonlinear and linearized part. Hence, let us denote δ := − l , m δ := m − m l , then they satisfy the following system with the zero initial data ( δ , m δ )(x, t)| t=0 = (0, 0). (2.20) Here, the force term is defined in (2.2). Now we will establish the decay rate for the solution ( δ , m δ ) of equation (2.19) in the following.
where l = 0, 1, 2, 3. Here, the energy E 3 l (t) is defined by Due to the smallness of δ, there are two constants C * and C * (independent of time) such that 26) where l = 0, 1, 2, 3. Here, C is a constant independent of time.
Proof. We will take the strategy of induction to give the proof of estimate (2.26). Taking l = 0 in (2.23), then we have d dt Obviously, the dissipation term ∇ δ 2 H 4 + ∇m δ 2 H 3 cannot control the energy term E 3 0 (t) in above inequality. Thus, we add both sides of the above inequality with term ( δ , m δ ) 2 L 2 and get d dt By virtue of the Duhamel principle formula and estimate (1.7), we have where we have used Sobolev inequality to get Using (2.27), (2.28) and equivalent relation (2.25), one arrives at It is easy to obtain that where we have used the fact that In the sequel, we only need to deal with the term We claim the estimate(which will be proved in Sect. 3), Hence, we can easily obtain that We now assume that the decay rate (2.26) holds on for the case k = l, i.e., (2.33) J. Gao, Z. Lyu and Z. Yao ZAMP For some constant R defined below, denoting the time sphere (see [19]) (2.34) By substituting (2.34) into (2.33), we can easily get . Choosing R = C * (l + 4)/C and multiplying the above inequality by (1 + t) l+4 , we obtain that We claim that the following estimate holds on (which will be proved in Sect. 3), Then, due to the fact that the term E 3 l+1 (t) is equivalent to the norm ∇ l+1 δ Thus, by the general step of induction, we have given the proof for (2.26).
Finally, we establish the lower bound estimates.
Proof. Recall the definition which, together with estimates (2.6) and (2.26), yields directly Choosing t ≥ 3 4 . Similarly, using estimates (2.6) and (2.26), we also have (2.38) Finally, we establish the lower decay rate for the velocity. Using decay estimate (1.7) and Morse inequality, we get which, together with (2.38), yields directly Therefore, we complete the proof of lemma.

Upper and lower bounds of decay for time derivative
In this subsection, we will establish the upper and lower bounds for the time derivatives of density and velocity. In [25], Wang and Tan have rewriten (1.1) in the perturbation form as The initial data are given as Now, we establish the upper and lower bounds of decay rate for the time derivative of solution in the L 2 norm. The lower decay rate estimate for the time derivative of density and velocity can be obtained by using the method in [9]. However, we still give the estimate in detail due to the appearance of the Korteweg term.
for all t ≥ t * with t * a positive constant. Here, C is a positive constant independent of time.
Proof. At first, we establish upper bound time decay rate for ∂ t and ∂ t u in the L 2 norm. With the help of the equation (2.39), we can easily obtain By virtue of Sobolev's inequality and time decay rate (1.7), we have and Then, we can easily derive that Next, we establish lower bound time decay rate for ∂ t u in the L 2 norm. Using the momentum equation in (2.39), we have With the help of the inequality (2.43), we can get for all t ≥ t * , Finally, we establish lower bound time decay rate for ∂ t in the L 2 norm. To achieve this target, we use the transport equation in (2.39) to obtain hence, together with the inequality (2.42), we obtain Now, we need to establish the lower bound decay rate for divu L 2 . Notice the differential relation Δ = ∇div − ∇ × ∇×, we get And hence, one arrives at Using Sobolev's inequality, uniform bound (1.6) and decay rate (1.7), we have By virtue of the Duhamel principle formula and (2.47), we get which, together with estimates (2.45) and (2.46), yields directly where the positive constant C on right-hand side of the above inequality depends on c 0 given in Proposition 2.1, but not on δ and δ 1 . Then, by virtue of the smallness of δ and δ 1 , we have for t ≥ t * , Therefore, we complete the proof of this lemma.

Proof of some technical estimates
In this section, we will establish the claim estimates that have been used in Sect. 2. That is to say, we will establish the claim estimates ( Proof of inequality (2.18). Let ζ = ξ √ t, we obtain |ζ|≤ηt 1 2 e −(μ+ν)|ζ| 2 |ζ| 2s cos 2 (|ζ|t where there exits a positive large time t * such that for t ≥ t * , we can obtain with C a positive constant that independent of time. Proof of inequality (2.21). Multiplying the first and second equations of (2.19) by δ and m δ , respectively, it holds on Due to the fact that P (1) = 1, we can use the Taylor expression formula to get which, together with Sobolev's inequality, yields directly where the symbol ∼ represents the equivalent relation. By virtue of integrating by parts and using the transport equation, it is easy to get Then, we get Applying the equation (2.19), it is easy to obtain for k = 1, 2, 3, At first, we estimate the term ∇ k+1 Δ δ · ∇ k m δ dx for k = 1, 2, 3. Using integration by parts and transport equation, we can obtain Now we give the estimates for ∇ k S 2 L 2 , k = 1, 2, 3. Indeed, when k = 1, we apply Sobolev's inequality to obtain Similarly, we also have for k = 1, By the same way, we have By virtue of the Taylor expression formula, we get where we have used the fact that P (1) = 1. Then, we use Sobolev's inequality to obtain Thus, it holds on for k = 1, Then, we use Cauchy inequality to get d dt Notice that and Next, employing Sobolev's inequality, it is easy to get Then, we obtain the following estimates . Therefore, we complete the proof of claim estimate (2.21).